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Sequence Domain Harmonic Modeling of Type-IV Wind Turbines

Guest, Emerson; Jensen, Kim Høj; Rasmussen, Tonny Wederberg

Published in:I E E E Transactions on Power Electronics

Link to article, DOI:10.1109/TPEL.2017.2735303

Publication date:2017

Document VersionPeer reviewed version

Link back to DTU Orbit

Citation (APA):Guest, E., Jensen, K. H., & Rasmussen, T. W. (2017). Sequence Domain Harmonic Modeling of Type-IV WindTurbines. I E E E Transactions on Power Electronics. DOI: 10.1109/TPEL.2017.2735303

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Sequence Domain Harmonic Modeling of Type-IVWind Turbines

Emerson Guest, Student Member, IEEE, Kim H. Jensen, and Tonny W. Rasmussen, Member, IEEE

Abstract—A sequence domain (SD) harmonic model of a grid-connected voltage-source converter is developed for decouplingconverter generated voltage harmonics from voltage harmonics inthe external grid. The modeling procedure includes a derivationof the baseband frequency response for regular-sampled pulse-width modulation and an analysis of converter generated voltageharmonics due to compensated dead-time. The decoupling ca-pabilities of the proposed the SD harmonic model are verifiedthrough a power quality (PQ) assessment of a 3MW Type-IVwind turbine. The assessment shows that the magnitude andphase of low-order odd converter generated voltage harmonicsare dependent on the converter operating point and the phaseof the fundamental component of converter current respectively.The SD harmonic model can be used to make PQ assessments ofType-IV wind turbines or incorporated into harmonic load flowsfor computation of PQ in wind power plants.

Index Terms—Power quality, Power conversion harmonics,Wind turbine, Pulse-width modulated power converters.

I. INTRODUCTION

THE assessment and computation of power quality (PQ)of grid-connected power-electronic devices is an ongoing

area of research for the energy industry [1]. The assessment ofPQ covers the testing of power electronic devices to quantifytheir harmonic emissions at different operating points. Thecomputation of PQ concerns calculating harmonic load flow inplanned or existing networks of power-electronic devices andpassive components. The computation of PQ in wind powerplants (WPPs) is normally undertaken in the planning phase.This is to ensure the proper allocation of passive filters suchthat the wind farm complies with harmonic emission limits.Obtaining a coherence between the models used to assess windturbine PQ and the models used to compute system PQ is acommon goal for wind turbine manufacturers and operators.

The standard approach for assessing the PQ of commercialwind turbines is based on IEC 61400-21:2008 [2] where ameasurement of the harmonic current injected by a windturbine is used to make a statement about its PQ. Currentmeasurements of DFIG wind turbines have been used to assessthe statistics of the magnitude and phase of current harmonics[3],[4], to assess harmonic propagation between wind turbines[5] and to quantify how active power production influencesthe magnitude of low-order harmonic current injection [6].Current source models derived from measurements have beenused to compute harmonic flows in a DFIG based wind farm

E. Guest and T. W. Rasmussen are with the Center for Electric Powerand Energy, Technical University of Denmark, Lyngby DK-2800, Denmark(e-mail: edagu@elektro.dtu.dk; twr@elektro.dtu.dk).

K. H. Jensen is with Siemens Gamesa Renewable Energy, Brande DK-7330,Denmark (e-mail: kim_hoej.jensen@siemens.com).

[7], to compute the aggregated current harmonics in a windfarm connected to a public grid [8] and the importance ofcorrectly modeling collector system when using current sourcemodels is identified in [9].

PQ computations based on current source models can beinaccurate due to a misrepresentation of the harmonic sources[1],[5],[10]-[11]. Consequently, a per-phase impedance-basedharmonic model of the grid-side voltage-source converter(VSC) of a Type-IV wind turbine was proposed in [12] to cal-culate the internal voltage source of the converter. Impedance-based models have also been used to study harmonic stabilityof grid-connected converters [13]-[15], resonance phenomenon[16]-[19], converter impedance shaping [20],[21] and har-monic propagation between converters and the grid [22],[23].Nonlinearities in dq frame current control have been shown tointroduce an unbalanced system within the bandwidth of thecurrent controller, precluding the application of single-inputsingle-output models [15],[24],[25]. One method for treatingthe unbalanced three-phase system is to transform the controlmatrices into the dq frame for analysis [26]. Alternatively,harmonic linearization can be employed [27] to transform thecontrol matrices into the sequence domain (SD).

Whilst impedance-based models rely on time averagingto rationalize the PWM process, more advanced frequencydomain approaches using harmonic transfer matrices (HTM)[28],[29] model the cross-frequency behavior within VSC con-trol and modulation systems. Alternatively, [30],[31] employthe so-called extended harmonic domain (EHD) to model theinter-modulation effects of PWM in dynamic simulations.

In this paper a SD harmonic model is proposed whichaugments the SD transfer matrix developed in [27] witha disturbance model to capture converter generated voltageharmonics due to dead-time [32]-[35] and the characteristicswitching harmonics [36],[37]. The SD harmonic model con-nects the assessment and computation of PQ by:

1) establishing a generic framework for decoupling con-verter generated voltage harmonics from voltage harmon-ics present in the external grid during the PQ assessment,

2) incorporating magnitude and phase information obtainedfrom the PQ assessment into the computation of systemPQ,

3) being tractable for PQ computation consisting of tens tohundreds of wind turbines.

The development of the SD harmonic model includes; ananalysis of converter generated voltage harmonics due tocompensated dead-time; and a derivation of the baseband fre-quency response of regular-sampled pulse-width modulation(PWM) to represent the inter-modulation effects in sampled

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PWM systems [38]. The decoupling capabilities of the SDharmonic model will be verified through a PQ assessment ofa 3MW Type-IV wind turbine. The results of the assessmentwill show that the magnitude and phase of low-order odd con-verter generated voltage harmonics are dependent on converteroperating point and the phase of the fundamental componentof converter current respectively.

The rest of the paper is organized as follows: the harmonicmodeling methods are developed in Section II, the case studyon a 3MW Type-IV wind turbine is covered in Section IIIand the paper is concluded in Section IV.

II. HARMONIC MODELING METHODOLOGY

A. Rationale for the Sequence Domain Harmonic Model

Table I shows the applicability of various harmonicmodeling methods to the assessment and computation ofPQ for grid-connected VSCs. The most complete time-domain representation of converters is through full-orderelectromagnetic transient (EMT) models. EMT modelssynthesize the exact PWM switching and control functionalityat the cost of modeling complexity and long simulation time.Cross-frequency models such as the EHD [30]-[31] or HTMs[28]-[29] can be applied to the computation of VSC PQbut are limited to idealized PWM models. Reduced-orderimpedance-based models have been applied to the studyof harmonic propagation between converters and the grid[22],[23], but are yet to consider converter generated voltageharmonics. Current source models, whilst based on realmeasurements [3]-[9], have shown poor accuracy whencomputing PQ in WPPs [1],[5],[10]-[11].

TABLE I: Comparison of harmonic modeling methods appli-cable to the assessment and computation of PQ. VH-very high,H-high, M-medium, L-low.

Method

Dom

ain

ofIm

plem

enta

tion

Bas

edon

Rea

lM

easu

rem

ent

App

licab

leto

PQA

sses

smen

t

App

licab

leto

PQC

ompu

tatio

n

Mod

elin

gC

ompl

exity

/Tim

e

EMT Time 7 3 3 VH

EHD [30]-[31] Time 7 7 3 H

HTM [28]-[29] Freq. 7 7 3 M

Current Inj. [3]-[9] Freq. 3 3 7 L

Imp. Based [22],[23] Freq. 7 7 7 L

Proposed Method Freq. 3 3 3 L

The proposed SD harmonic model, developed in Section II-B,uses an internal representation of the converter harmonicsource to decouple converter generated voltage harmonicsfrom voltage harmonics in the external grid present in thePQ assessment. The ascertained SD harmonic model can thenbe integrated into harmonic load flows for PQ computations,as required in the planning of WPPs and potentially otherrenewable energy installations.

B. Modeling Procedure

A block diagram of a three-phase grid-connected VSC isshown in Fig. 1a. iabc(t) and vabc(t) are time-domain vectorsof the converter line currents and phase-ground voltages atthe grid-side of the line inductor L respectively. vlabc(t) is thevector of converter load-voltages, vmabc(t) is the modulatingsignal to the PWM block and s(t) is the vector of gate signalsto the VSC. The linearized converter model is shown in Fig.1b for a generic stationary or dq frame current controller withvoltage feed-forward such that,

vmabc(s) = Gabcvabc(s) + Habciabc(s). (1)

Gabc and Habc are 3 × 3 transfer matrices of the volt-age feed-forward and current control respectively, includingcomputational and sampling delays. Additionally, the linearpart of the modulation process is modeled by a basebandfrequency response, Kabc, and the nonlinear part modeledas a disturbance vdsabc(t). The disturbance model isolates theconverter generated voltage harmonics due to dead-time error-pulses [32]-[35] and the characteristic sideband harmonics[36],[37]. In general the spectrum of vdsabc(t) is dependent onthe converter operating point, modulation scheme and to alesser degree, background harmonics in the grid. The blockdiagram in Fig. 1b is transformed into the SD by the sequencecomponent decomposition, T, defined with respect to the a-phase, and the substitution s = jω,

vdspn0(jω) =(I− TKabcGabcT−1)vpn0(jω)+

(−TKabcHabcT−1 + jωLI)ipn0(jω).(2)

The transfer matrices Kpn0Hpn0 = TKabcHabcT−1 andKpn0Gpn0 = TKabcGabcT−1 are the current controller andvoltage feed-forward referred to the sequence domain. I is theidentity matrix. Omitting zero-sequence components and re-arranging (2) gives,

vdpn(jω) = Zipn(jω) + vpn(jω), (3)

with the SD converter impedance,

Z = (I−KpnGpn)−1(−KpnHpn + jωLI), (4)

and apparent harmonic voltage source generated by the con-verter,

vdpn(jω) = (I−KpnGpn)−1vdspn(jω). (5)

Note that vdpn(jω) is acted upon by the voltage feed-forwardand thus cannot be directly measured at the converter termi-nals. Z has the form,

Z =

[Zpp(jω) Zpn(jω)Znp(jω) Znn(jω)

]. (6)

Under balanced and slightly unbalanced power system oper-ation Zpn(jω) = Znp(jω) ≈ 0 [27]. In general, Zpp(jω) 6=Znn(jω) for dq frame current controllers within the band-width of the current controller [24]-[27] and for stationaryframe current control when impedance shaping techniques fordamping of power system current or voltage harmonics are

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dcV

L

Control

v )(tabci )(tabc)(ts

)(tmabcv

)(tlabcv

PWM

(a) Three-phase grid-connected VSC.

v )(tabc

i )(tabc

)(tmabcv

Kabc

Habc

Gabc

sL

1

PWM + Converter

)(tlabcv

)(tds

abcv

(b) Linearized VSC model containing a PWM model,disturbance model and control transfer functions.

)( 1jhZ pp

)( 1jhvd

p

)( 1jhvd

n

)( 1jhin

)( 1jhip

)( 1jhZnn)( 1jhvn

)( 1jhvp

Positive

Sequence

Network

Sequence Domain Harmonic Model

Negative

Sequence

Network

(c) SD harmonic model of VSC connected to an arbitrary linearnetwork for PQ computation.

Fig. 1: Development of the SD harmonic model for theassessment and computation of PQ.

used [21]. The SD harmonic model is obtained by evaluating(3) at harmonic frequencies ω = hω1 giving,

vdpn(jhω1) =[Zpp(jhω1) 0

0 Znn(jhω1)

]ipn(jhω1) + vpn(jhω1),

(7)where ω1 is the fundamental frequency of the power systemand h = 2, . . . , hmax are the harmonic orders of interest.Equation (7) is depicted in Fig. 1c as two decoupled Theveninequivalent circuits, one set for the positive sequence andnegative sequence harmonic components respectively. Thisis a generalization of the harmonic model proposed in [12]which is based on a per-phase equivalent circuit. The needfor a SD harmonic model becomes clear when Zpp(jhω1) 6=Znn(jhω1) or vdpn(jω) contains unbalanced harmonic phasors.Both situations can occur in a typical grid-connected converterapplication.

C. Baseband Frequency Response of a Regular-SampledPulse-Width Modulator

Harmonics enter the converter control paths due to theexistence of both converter generated and grid voltage har-monics. Therefore, the converter modulating signals, vmabc(t),are in general multi-tone signals consisting of a fundamentaland harmonic components. Regular-sampled PWM subjectto a multi-tone modulating signal cannot be treated as alinear amplifier due to inter-modulation in the baseband range[38]. The continuously-sampled zero-order hold (ZOH) withfrequency response,

KZOH(jω) = sinc(ωTs/2)e−jωTs/2, (8)

is sometimes used to model the low-frequency behavior ofthe PWM process [14],[27] but does not consider inter-modulation effects. Consequently, a baseband frequency re-sponse of regular-sampled PWM, KM (jω), is derived inthe Appendix which for asymmetrical regular-sampled PWMgives,

KM (jω) = J0(ωM1Ts/2)e−jωTs2. (9)

M1 is the magnitude of the fundamental component in themodulating signal, Ts is the sampling period and J0(z) isthe 0-th order Bessel function of the 1st kind. Note that (9)is a function of M1 which accounts for the inter-modulationbetween the fundamental and harmonic components in themodulating signal. As M1 ≈ 1 for grid-connected VSCsthe difference between (8) and (9) is the roll-off of sinc(z)compared to J0(z).

Equation (9) was verified by a PSCAD (EMTDC) switchingmodel of a two-level VSC. The model used a constant 1100VDC link voltage, a pulse ratio p = ωc/ω1 = 50, a balancedthree-phase passive load and a modulation index1 set atM1 = 1. The small-signal gain of the normalized output loadvoltage to the input modulating signal was measured. A 5µssimulation time-step was found to be sufficient for model ac-curacy. Fig. 2 shows how |KM (jω)| and |KZOH(jω)| divergewith frequency. This suggests the importance of using theactual frequency response of the modulator given in (9), whenthe calculation of higher frequency harmonic components gen-erated by the converter are of interest. Furthermore, the exactbaseband response of the converter may improve impedanceshaping techniques that rely on algebraic approximation ofconverter impedance, particularly for shaping in the midbandrange [21].

Harmonics in the modulating signal due to converter andgrid voltage harmonics also produce cross-frequency andcross-sequence harmonic sideband responses in the converterload voltages. The cross-frequency and cross-sequence prop-erties of the harmonic sideband responses preclude linearsystem modeling (but could be captured with a HTM similarto those used in [28]-[31]). Fortunately, the harmonic sideband

1A triplen third-harmonic component of magnitude M1/6 was also addedto the modulating signal to extend the modulator’s linear range to 1.15. Strictlyspeaking a factor of J0(ωM1Tc

24) due to the third harmonic component should

be added to (9), however, it has negligible impact on KM (jω) over thefrequency range of interest.

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10 20 30 40 50Harmonic Order (h)

0

0.2

0.4

0.6

0.8

1

| · |

|KZOH(jω)| - Analysis

|KM (jω)| - Analysis

|KM (jω)| - EMTDC

Fig. 2: Modulator gains for asymmetrically regular-sampledPWM and continuously-sampled ZOH when p = ωc/ω1 = 50.

responses are considerably smaller in magnitude than the base-band response and are thereby ignored for a significant modelsimplification2. When the SD harmonic model is applied toa VSC the harmonic sideband responses are captured in theapparent harmonic voltage source, vdpn(jω).

D. Deriving the Converter Impedance

The converter impedance, Z, required for the case studyon a Type-IV wind turbine in Section III is now used toexemplify the deviation that can occur between SD converterimpedances. Z is calculated using (4) and knowledge of theVSC specific control and modulation transfer matrices Gpn,Hpn (not provided herein) and Kpn (see (19) in the appendix)respectively. The analytical converter impedance is verifiedagainst an empirical result extracted by applying a harmonicperturbation [40] to an EMTDC model3 of the wind turbineelectrical drive-train, control software implementation and gridconnection. The deviation between |Zpp(jω)| and |Znn(jω)|in the low-harmonic range, shown in Fig. 3, aligns with similaranalyses found in [18],[27]. Resonances occurring in the low-harmonic range are found in WPPs [18],[19] and systemsconnected through long AC cables [16],[17]. The differencebetween |Zpp(jω)| and |Znn(jω)| and a system resonancein the low-harmonic range can cause sequence dependentharmonic current amplification [27].

E. Magnitude and Phase of Converter Generated Odd VoltageHarmonics due to Dead-time Error Pulses

The apparent harmonic voltage source, vdpn(jhω1), capturesthe converter generated voltage harmonics. The characteris-tic sideband harmonics that appear around multiples of theswitching frequency, fc, are primarily defined by the modula-tion scheme and the fundamental component in the modulationsignal [36],[37]. Within the linear modulating range, low-order

2Closed-form expressions for the harmonic sideband responses can bederived from (14) and (15) in the Appendix.

3In reality the high-power converter introduces low-order voltage harmonicsthat can distort the converter impedance measurement. The simulation envi-ronment has an advantage because nonlinear PWM effects can be minimized.

2 4 6 8 10 12 14Harmonic Order (h)

10-2

10-1

|Ω| Zpp

- Analysis

Znn

- Analysis

Zpp

- EMTDC

Znn

- EMTDC

Fig. 3: Magnitude of converter impedances, |Zpp(jω)| and|Znn(jω)|, calculated analytically and extracted empiricallyfrom a simulation model.

harmonics (f ≤ fc/2) are predominately contributed by dead-time [32],[35]. In high-power applications the dead-time delay,Td, is typically in the range of 1% − 3% of the switchingperiod, Tc.

In uncompensated dead-time (UDT) the rising edge on eachswitching device is delayed by Td and during the dead-timeinterval the converter terminal voltage clamps to either the+Vdc or −Vdc rail depending on the polarity of the loadcurrent. If the clamped voltage differs from the ideal voltagethen an error pulse of height Vdc and width Td occurs. Theerror pulses due to UDT and sinusoidal load currents havebeen shown using averaging [32]-[34] to approximate a squarewave with complex Fourier coefficients,

Uh =2VdcTdjhπTc

. (10)

The harmonic generating effects of compensated dead-time(CDT), however, are relatively undocumented. Therefore, acloser look will be given to CDT as it is often incorporatedinto high-power converters such as that found in the 3MWType-IV wind turbine studied in Section III. In CDT theactive switching edge is either delayed or advanced by Tddepending on the polarity of the sampled converter current[33]. With sinusoidal load currents the resultant switchingedges are theoretically ideally placed. In practice the converterload currents are not sinusoidal and can exhibit multiple zero-crossings due to a superposition of the switching ripple and thefundamental component. The multiple zero-crossings of theload current can cause the dead-time to be misplaced resultingin error pulses. This is shown in Fig. 4 for the a-phase risingcurrent, ia(t), where the polarity of the current, sgn(ia(t)),changes sign after sampling but before the switching edge. Thedead-time is placed on the wrong switching device resulting inan incorrect phase-neutral terminal voltage vlan(t) and a non-zero error pulse signal ea(t). The resultant train of error pulsestend to have the same polarity as the fundamental load current,sgn(ia,1(t)). This is opposite to what occurs in UDT wherethe error pulses have the opposite polarity to the fundamentalload current [32]-[34]. Moreover, the error pulses due to CDTare centered around the fundamental current zero-crossing.

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)(tia 1)(sgn tia 1)(sgn tia

)(tea

)(tvlan

dcV

dcV dT t

)(1, tia

2/cT

Advanced ‘off’ edge Delayed ‘on’ edge

1)(sgn tia

Fig. 4: Generation of the a-phase to neutral error pulse signal,ea(t), due to the load current, ia(t), changing polarity aftersampling but before the switching edge.

A challenge with modeling CDT is that the placement of theerror pulses is a function of both the instantaneous switchingtime of the PWM scheme and the polarity of the load currentresponse. EMTDC simulations can handle such nonlinearitybecause the system matrices are updated at each time-step.Cross-frequency models such as HTMs [28]-[29] or the EHD[30]-[31] cannot, at this time, handle such nonlinearity. Asimplified model is obtained by assuming an equal numberof error pulses kd ∈ N occur either side of the fundamentalcurrent zero-crossing. Averaging over each switching period,Tc, gives the averaged error pulse signal, ea(t), with averagederror pulses of height VdcD, where D = Td/Tc, and widthkdTc as shown in Fig. 5. The phase of ea(t) is determinedby the phase of the fundamental component of current, θi,1.The averaged error pulse signals for the b- and c-phases areobtained by a −120 and −240 phase shift respectively.

t

)(tea

cdTk

)(1, tia

dcDV

Fig. 5: Averaged a-phase to neutral error pulse signal, ea(t),for kd error pulses either side of the a-phase fundamentalcurrent zero-crossing.

For balanced load currents, eabc(t) will be balanced andproduce characteristic power system harmonics [1] where allnon-zero positive and negative sequence odd harmonics haveorder h = 6k + 1 and h = 6k − 1 respectively, where k ∈ N.From Fig. 5 the complex Fourier coefficients, Ch, of ea(t) are,

Ch =ω1

2π

∫ T1+θi,1/ω1

θi,1/ω1

ea(t)e−jhω1tdt

=2VdcD

hπ

[1− cos

(kd

2πh

p

)]ejhθi,1 ,

(11)

where p = ωc/ω1 is the pulse ratio and h is odd. Theactive/reactive power injection (converter operating point) ina wind turbine will determine the magnitude of the funda-mental load currents and influence the number of currentzero-crossings. Consequently, (11), being a function of kd,indicates that the low-order voltage harmonics generated by

the converter due to error pulses will vary with the converteroperating point. Furthermore, the phase of the error pulseharmonics are seen to be linearly related to the phase of thefundamental component of current as hθi,1.

Fig. 6 shows the magnitude spectrum4, 2|Ch|, for differentvalues of kd when D = 0.01, Vdc = 1100 and p = 50. Thelargest spectral components appear at harmonic multiples ofthe fundamental. This aligns with the conventional notion ofusing CDT to improve converter performance at the fundamen-tal frequency [32],[33]. It is apparent that the harmonic orderwith the largest magnitude moves closer to the fundamentalas kd increases. This indicates that lower-order harmonics willtend to increase with decreasing active power injection. Alsoshown in Fig. 6 is the envelope of Ch calculated as,

maxkd

(|Ch|) =4VdcD

hπ= 2 · |Uh|. (12)

Equation (12) indicates that across a range of converter oper-ating points (values of kd) the envelope of Ch is twice that ofUh given in (10). Therefore, CDT can introduce harmonics ofsignificantly larger magnitude than UDT.

5 10 15 20 25Harmonic order (h)

0

1

2

3

4

5

|V |

2|Ch|, kd = 1

2|Ch|, kd = 2

2|Ch|, kd = 3

maxkd

(2 · |Ch|)

Fig. 6: Magnitude of voltage harmonics due to averaged errorpulses, 2|Ch|, for different operating points (values of kd).

In general there will not be identically kd error pulseseither side of the fundamental zero crossing. In some cases anerror pulse will be dropped or an error pulse will be added,resulting in an unbalanced three-phase signal and therebyintroducing even and non-characteristic odd harmonics. Theprecise arrangement and quantity of the error pulses is afunction of the phase of the PWM carrier wave, φc.

III. CASE STUDY - SEQUENCE DOMAIN HARMONICMODEL OF A 3MW TYPE-IV WIND TURBINE

Two SD harmonic model instances were obtained for a3MW Type-IV wind turbine converter with the ratings givenin Table II. The first instance was derived from voltage andcurrent measurements of the actual wind turbine connectedto the Danish power grid. The instance was derived fromsimulated data taken from an EMTDC model of the windturbine, connected to a lumped grid model (short-circuit

4The factor 2 is required to obtain the magnitude of the real Fouriercoefficients.

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ratio of 10) containing low-order voltage harmonics accruing1.5% total harmonic distortion and representing a mildlydistorted grid [39]. The EMTDC model consisted of reduced-order mechanical model and a switching model for theconverter including the control software and the dead-timecompensation scheme. The grid connection in the EMTDCmodel was intentionally chosen to differ from the measuredgrid. This is to emphasize the decoupling capabilities of theSD harmonic model under different grid conditions.

TABLE II: Ratings of the Type-IV wind turbine grid-side VSC.

Parameter Value SymbolRated Active Power 3 MW PratedRated Frequency 50 Hz f1

Rated Voltage (line-line) 690 VRMS -Switching Frequency 2.5 kHz fcSampling Frequency 5 kHz fsDead-time Delay 4 µs TdDead-time Scheme CDT -

A. Harmonic Measurement Procedure

The harmonic measurement procedure applied to the windturbine was based on [2] and is summarized as follows:

1) Set the first operating point as P = 0.1 · Prated , Q ≈ 0,where P , Q are the active and reactive power injectionsrespectively.

2) Sample vabc(t), iabc(t) (see Fig. 1a) at 20kHz for 600s.3) Repeat step 2) for the remaining operating points i.e. P =

0.2 · Prated, . . . , 1 · Prated , Q ≈ 0.4) Use the fundamental zero-crossings of ia(t) to partition

samples into blocks of data corresponding to 10 funda-mental cycles (12 cycles for 60Hz systems).

5) Calculate the FFT for each block of samples to obtainthe SD phasors vpn(jhω1) and ipn(jhω1).

6) Calculate vdpn(jhω1) with (7).

B. Decoupling of the Apparent Harmonic Voltage Source forP = 0.5 · Prated

The 5th and 7th harmonic current phasors, in(j5ω1) andip(j7ω1), and apparent harmonic voltage phasors, vdn(j5ω1)and vdp(j7ω1), for each block of data outputted by the FFTwere chosen to demonstrate the decoupling capabilities of theSD harmonic model when P = 0.5 ·Prated. Fig. 7 shows thatthe EMTDC and measured data for in(j5ω1) and ip(j7ω1)differ in both amplitude and phase, which is expected becausethe current response depends on the converter voltage, thegrid voltage and the interconnecting impedance. The EMTDCand measured data for vdn(j5ω1) and vdp(j7ω1), however, areapproximately equal in both magnitude and phase. Moreover,their phase is very close to 0. This is expected because thefundamental current zero-crossing was used to synchronizethe sampling, i.e. θi,1 ≈ 0, and the phase of the low-orderharmonics due to dead-time error pulses was shown in SectionII-E to be hθi,1, therefore, hθi,1 ≈ 0.

In contrast to the phase-locked harmonics shown in Fig.7, the fixed frequency carrier sideband harmonic vdp(j48ω1),

Fig. 7: Phasor diagrams for the converter currents in(j5ω1),ip(j7ω1) and apparent harmonic voltage sources vdn(j5ω1),vdp(j7ω1) when P = 0.5 ·Prated. EMTDC data (), measureddata (×).

shown in Fig. 8, exhibits uniformly distributed phase over[−180 180]. The phase distribution is because the gridfrequency, estimated as f1 = 49.9625Hz, causes a frequencyerror between the 48th bin of the FFT, located at 48f1 =2398.2Hz, and the fixed frequency of the sideband harmonic,located at fc − 2f1 = 2400.075Hz. It was mentioned inSection II-E that non-symmetry in the dead-time error pulseswould introduce even and non-characteristic odd harmonicsthat are a function of PWM carrier phase φc. An example ofthis is shown in Fig. 8 for vdp(j6ω1) which also exhibits uni-formly distributed phase over [−180 180]. In fact, all evenand non-characteristic odd harmonics at all operating pointstend to exhibit this phase distribution. A similar observationis noted in [11].

-100 0 100Re·

-150

-100

-50

0

50

100

150vdp(j48ω1)

Im·

-0.2 0 0.2Re·

-0.3

-0.2

-0.1

0

0.1

0.2

vdp(j6ω1)

Fig. 8: Phasor diagrams for the apparent harmonic voltagesources vdp(j48ω1) and vdp(j6ω1) when P = 0.5 · Prated.EMTDC data (), measured data (×).

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C. Magnitude and Phase of the Apparent Harmonic VoltageSource vs Active Power

Section II-E identified that characteristic harmonics due todead-time error pulses will express themselves in vdpn(jhω1)as low-order positive and negative sequence harmonics oforder h = 6k + 1 and h = 6k − 1 respectively, where k ∈ N.Fig. 9 shows how the magnitudes, |vdn(j(6k − 1)ω1)| and|vdp(j(6k + 1)ω1)|5 for 1 ≤ k ≤ 4, vary with active powerinjection. The alignment between the EMTDC and measureddata reinforces how a common SD harmonic model of theconverter can be extracted regardless of the external grid con-ditions. This is not possible using only current measurementas shown in Section III-B. Fig. 6 indicated that |vdn(j5ω1)|and |vdp(j7ω1)| should increase for increasing kd (decreasingactive power) which aligns with Fig. 9. Similarly, Fig. 6indicated that |vdn(j11ω1)| and |vdp(j13ω1)| should have theirlargest magnitude at a lower value of kd, also reflected inFig. 9. Similar conclusions can be drawn for |vdn(j17ω1)| to|vdp(j25ω1)|.

0 0.5 10

2

4 |v dn (j5ω1)|

|V |

EMTDC

Measured

0 0.5 10

2

4

|v dp (j7ω1)|

0 0.5 10

1

2

3

|v dn (j11ω1)|

|V |

0 0.5 10

1

2 |v dp (j13ω1)|

0 0.5 10

0.5

1

1.5|v d

n (j17ω1)|

|V |

0 0.5 10

0.5

1

1.5|v d

p (j19ω1)|

0 0.5 1Active Power (pu)

0

0.5

1

1.5|v d

n (j23ω1)|

|V |

0 0.5 1Active Power (pu)

0

0.5

1

1.5|v d

p (j25ω1)|

Fig. 9: Magnitude of the apparent harmonic voltage source,|vdn(j(6k − 1)ω1)| and |vdp(j(6k + 1)ω1)| for 1 ≤ k ≤ 4, as afunction of active power injection.

Fig. 10 shows how the phases, arg(vdn(j(6k − 1)ω1)) andarg(vdp(j(6k + 1)ω1)) for 1 ≤ k ≤ 4, vary with active powerinjection. It was commented in Section III-B that as θi,1 ≈ 0

the phases of the low-order harmonics due to dead-timeshould be hθi,1 ≈ 0. Fig. 10 shows that arg(vdn(j5ω1)) to

5To improve clarity only the mean value of amplitude is plotted.

arg(vdp(j13ω1)) remain close to 0 across all operating points(with some deviation occurring for lower active power bins).The phases of the higher harmonic orders, arg(vdn(j17ω1))to arg(vdp(j25ω1)), remain approximately in the first quadrant[−45 45]. The deviations between the phases obtained fromthe EMTDC and measured data can result from unmodeledPWM behavior, differences in the PWM carrier phase, anddifferences in the instantaneous sampling of the voltages andcurrents used to calculate vdpn(jhω1) in the two modelinginstances.

0 0.5 1

-45

0

45

arg(v dn (j5ω1))

degEMTDC

Measured

0 0.5 1

-45

0

45arg(v d

p (j7ω1))

0 0.5 1

-45

0

45arg(v d

n (j11ω1))deg

0 0.5 1

-45

0

45arg(v d

p (j13ω1))

0 0.5 1

-45

0

45

90arg(v d

n (j17ω1))deg

0 0.5 1

-45

0

45

90arg(v d

p (j19ω1))

0 0.5 1Active Power (pu)

-45

0

45

90arg(v d

n (j23ω1))deg

0 0.5 1Active Power (pu)

-45

0

45

90arg(v d

p (j25ω1))

Fig. 10: Phase of the apparent harmonic voltage source,arg(vdn(j(6k−1)ω1)) and arg(vdp(j(6k+1)ω1)) for 1 ≤ k ≤ 4,as a function of active power injection.

Fig. 11 shows a virtual spectrum of the worst-case low-order harmonics obtained by taking the maximal elements of|vdpn(jhω1)| from each operating point. Overlaid on Fig. 11 isthe envelope of the averaged CDT model, max

kd(2 · |Ch|) de-

rived in (12), showing that the worst-case low-order harmonicsobtained from EMTDC and measurement are bounded fromabove as expected. Moreover, comparing Fig. 11 with Fig. 9highlights the importance of assessing PQ across a range ofoperating points as the operating point for which the largestharmonic magnitude occurs differs for each harmonic order.

D. A Comment on the Aggregation of Current Harmonics inWind Farms Consisting of Type-IV Wind Turbines

The narrow range of the phase of the characteristic low-order odd voltage harmonics generated by the converter, exhib-ited in Fig. 10, has an important bearing on WPPs consisting of

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5 10 15 20 25 30Harmonic Order (h)

1

2

3

4

5

|V |

EMTDC max(

|v dpn(jhω1)|

)

Measured max(

|v dpn(jhω1)|

)

Averaged CDT maxkd

(2 · |Ch|)

Fig. 11: Virtual spectrum of the maximal elements of|vdpn(jhω1)| taken across all active power injections.

Type-IV wind turbines. Each wind turbine synchronizes to thegrid voltage and injects a fundamental component of currentto meet the active power requirement. The small voltage dropbetween each wind turbine in the collector system results inan approximately in-phase fundamental current injection byeach wind turbine. Based on the findings in Section III-C, theconverter generated odd voltage harmonics (at least for theassessed wind turbine) will tend to be in-phase, even acrossdifferent active power injections, and thus inject harmoniccurrents that aggregate constructively at the point of commoncoupling. Accordingly, one would not expect the Rayleightype attenuation of magnitude in the aggregated current, asdiscussed in [7], for these particular harmonics. Other windturbines with different control or modulation strategies mayexhibit different phase characteristics for low-order odd har-monics. These characteristics can likewise be extracted usingthe SD harmonic model.

IV. CONCLUSION

A SD harmonic model of grid-connected VSCs has beenintroduced for decoupling converter generated voltage har-monics from voltage harmonics in the external grid. The SDharmonic model provides a general solution for assessing andcomputing PQ in individual and systems of Type-IV windturbines respectively. The development of the SD harmonicmodel included a derivation of a baseband frequency responsefor regular-sampled PWM.

The SD harmonic model was used to assess the PQ of a3MW Type-IV wind turbine. The assessment showed that themagnitude of low-order converter generated odd voltage har-monics were dependent on the converter operating point. Thisaligned with an analysis of error pulses due to compensateddead-time obtained through an averaged value model. Thephase of the aforementioned voltage harmonics were shownto be dependent on the phase of the fundamental componentof current. This should be considered when computing PQ forWPPs consisting of Type-IV wind turbines.

APPENDIX - BASEBAND FREQUENCY RESPONSE OFREGULAR-SAMPLED PULSE WIDTH MODULATION

Consider a normalized three-phase modulating signal,vmabc(t), with a fundamental component of positive phase

sequence and harmonic components of positive and negativephase sequence,

vmabc(t) = vabc,1(t) + vabc,p(t) + vabc,n(t)

= M1

[cos(ω1t+φ1)

cos(ω1t−2π/3+φ1)cos(ω1t+2π/3+φ1)

]+

N∑k=2

Mp,k

[cos(kω1t+φp,k)

cos(kω1t−2π/3+φp,k)

cos(kω1t+2π/3+φp,k)

]

+N∑k=2

Mn,k

[cos(kω1t+φn,k)

cos(kω1t+2π/3+φn,k)

cos(kω1t−2π/3+φn,k)

],

(13)where M1, φ1, Mp,k, φp,k and Mn,k, φn,k are themagnitude and phase of the positive sequence fundamental,positive sequence harmonic and negative sequence harmoniccomponents respectively, k is the harmonic order. The mod-ulating signal is applied to a two-level VSC using a regular-sampled PWM scheme and connected to a balanced three-phase load. The Poisson re-summation method proposed in[37] is applied to (13) giving the SD load voltage spectra,

vlpn(ω) =∑m,n

am,n3

[1+2 cos( 2π

3 (x1−1+∑Nk=2(xp,k−xn,k)))

1+2 cos( 2π3 (x1+1+

∑Nk=2(xp,k−xn,k)))

]· 2πδ(ω − Ωmn).

(14)where am,n are the complex Fourier coefficients,

am,n =4

jΩmnTcJx1(ΩmnM1Tc

4 )ej(x1φ1+mφc)·

N∏k=2

Jxp,k(ΩmnMp,kTc

4 )ejxp,kφp,kJxn,k(ΩmnMn,kTc

4 )ejxn,kφn,k ·

j(m+n)) cos(nω1Tc4 (1−R))e−jnω1

Tc4 (2−R).

(15)Tc is the switching period, δ(ω − Ωmn) is the delta functioncentered at ω = Ωmn, Ωmn = nω1 + mωc is the frequencycomponent for the mth and nth indexes, −π ≤ φc ≤ π isthe phase of the PWM carrier and R is a binary parameterwith R = 0 for symmetrically sampled PWM and R = 1for asymmetrically sampled PWM. Additionally, n = x1 +∑Nk=2(kxp,k+kxn,k) where x1, xp,k, and xn,k are the indexes

of the fundamental, k-th order positive sequence harmonicand k-th order negative sequence harmonic components in themodulating signal respectively. The baseband response for theh-th order positive sequence harmonic occurs for the indexesm = 0, x1 = 0, xp,h = 1, xp,k = 0 ∀ k 6= h, xn,k = 0 ∀ ksuch that Ωmn = hw1 and,

vlpn(ω) =4

hω1TcJ0(hω1

M1Tc4 )J1(hω1

Mp,hTc4 )ejφp,h ·

N∏k=2k 6=h

J0(hω1Mp,kTc

4 )N∏k=2

J0(hω1Mn,kTc

4 )·

cos(hω1Tc4 (1−R))e−jhω1

Tc4 (2−R)

[10

]· 2πδ(ω − hω1).

(16)Consider (16) for asymmetrically regular-sampled PWMwhere R = 1 and ωs = 2ωc. The simplification J0(z) ≈ 1is within 1% error if z ≤ 0.2. If z = hω1Mx,hTc/4, wherex = p, n, then Mx,h ≤ 0.1273 · q/h where q = ωc/ω1 isthe pulse ratio. If the modulating signal is band-limited to

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qω1 then Mx,q ≤ 0.1273 is required for the simplificationto hold. Similarly, the Bessel function J1(z) ≈ z/2 whenz 1. If z = hω1Mp,hTc/4 and h = q then J1(z) ≈ z/2 iswithin 1% error if Mp,h ≤ 0.63. The loads supplied by powerconverters typically have low-pass impedance and convertercontrol systems are normally designed to resonate at thefundamental frequency and dampen at harmonic frequencies.Therefore, under normal operation Mx,h 0.1, and (16)simplifies to,

vlpn(ω) = J0(hw1M1Tc

4 ) cos(hω1Tc4 (1−R))e−jhω1

Tc4 (2−R)·

Mp,h

2ejφp,h

[10

]· 2πδ(ω − hw1).

(17)Similarly, it can be shown that the main baseband responsefor the h-th order negative sequence harmonic component inthe modulating signal is given by,

vlpn(ω) = J0(hw1M1Tc

4 ) cos(hω1Tc4 (1−R))e−jhω1

Tc4 (2−R)

· Mn,h

2ejφn,h

[01

]· 2πδ(ω − hw1).

(18)Equations (17) and (18) can be considered as the output har-monic phasors to the input harmonic phasors Mp,he

jφp,h andMn,he

jφn,h respectively. The transfer matrix representation forthe SD baseband frequency response is then,

Kpn =[KM (jω) 0

0 KM (jω)

], (19)

where,

KM (jω) = J0(ωM1Tc4 ) cos(ω Tc4 (1−R))e−jω

Tc4 (2−R).

(20)

ACKNOWLEDGMENT

This work was supported in part by Siemens GamesaRenewable Energy (SGRE) and in part by the InnovationsFund, Denmark. The authors thank P. Brogan, L. Shuai andR. Sharma of SGRE and Professors V. Agelidis and J. Holbøllof DTU for their valuable input and discussions.

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Emerson Guest received the Bachelor degree(2010) from the University of Newcastle, Newcas-tle, Australia, and the M.Sc. degree (2014) fromthe Technical University of Denmark, Kgs. Lyngby,Denmark, where he is currently working towards theIndustrial Ph.D. degree, all in electrical engineering.

His research interests include harmonic modelingof grid-connected converters, active filtering andreal-time optimization of power electronic systems.

Kim H. Jensen holds M.Sc. (1999) and Ph.D.(2003) degrees from the Technical University ofDenmark, Kgs. Lyngby, Denmark. He has beenworking in a transmission company (NESA), consul-tant company (Elsam Engineering) and now SiemensGamesa Renewable Energy. During his career he hasworked with transmission system planning, networkdesigns, wind turbine modeling, grid code studies,harmonic analysis and stability studies. He has alsoworked as a technical specialist on a number of windfarm integration projects.

Tonny W. Rasmussen holds M.Sc. (1993) andIndustrial Ph.D. (1998) degrees from the TechnicalUniversity of Denmark, Kgs. Lyngby, Denmark.He is an Associate Professor for the Centre forElectric Power and Energy, Technical University ofDenmark, Kgs. Lyngby, and a Member of the IEEEPower Electronics Society.

His research interests include power electronicconverter topologies, converter control and grid-connected applications of converters.